statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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lower_crossing_time_lt_of_lt_upcrossings_before
(hN : 0 < N) (hab : a < b) (hn : n < upcrossings_before a b f N ω) :
lower_crossing_time a b f N n ω < N | lt_of_le_of_lt lower_crossing_time_le_upper_crossing_time_succ
(upper_crossing_time_lt_of_le_upcrossings_before hN hab hn) | lemma | measure_theory.lower_crossing_time_lt_of_lt_upcrossings_before | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sub_of_le_upcrossings_before
(hN : 0 < N) (hab : a < b) (hn : n < upcrossings_before a b f N ω) :
b - a ≤
stopped_value f (upper_crossing_time a b f N (n + 1)) ω -
stopped_value f (lower_crossing_time a b f N n) ω | sub_le_sub (stopped_value_upper_crossing_time
(upper_crossing_time_lt_of_le_upcrossings_before hN hab hn).ne)
(stopped_value_lower_crossing_time (lower_crossing_time_lt_of_lt_upcrossings_before hN hab hn).ne) | lemma | measure_theory.le_sub_of_le_upcrossings_before | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_eq_zero_of_upcrossings_before_lt (hab : a < b) (hn : upcrossings_before a b f N ω < n) :
stopped_value f (upper_crossing_time a b f N (n + 1)) ω -
stopped_value f (lower_crossing_time a b f N n) ω = 0 | begin
have : N ≤ upper_crossing_time a b f N n ω,
{ rw upcrossings_before at hn,
rw ← not_lt,
exact λ h, not_le.2 hn (le_cSup (upper_crossing_time_lt_bdd_above hab) h) },
simp [stopped_value, upper_crossing_time_stabilize' (nat.le_succ n) this,
lower_crossing_time_stabilize' le_rfl
(le_trans thi... | lemma | measure_theory.sub_eq_zero_of_upcrossings_before_lt | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"le_cSup",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_upcrossings_before_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossings_before a b f N ω ≤
∑ k in finset.range N, upcrossing_strat a b f N k ω * (f (k + 1) - f k) ω | begin
classical,
by_cases hN : N = 0,
{ simp [hN] },
simp_rw [upcrossing_strat, finset.sum_mul, ← set.indicator_mul_left, pi.one_apply,
pi.sub_apply, one_mul],
rw finset.sum_comm,
have h₁ : ∀ k, ∑ n in finset.range N,
(set.Ico (lower_crossing_time a b f N k ω) (upper_crossing_time a b f N (k + 1) ω)... | lemma | measure_theory.mul_upcrossings_before_le | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"and_iff_right_iff_imp",
"and_imp",
"finset.Ico",
"finset.card_range",
"finset.filter",
"finset.mem_Ico",
"finset.mem_filter",
"finset.mem_range",
"finset.range",
"finset.sum_mul",
"mul_comm",
"nsmul_eq_mul",
"one_mul",
"pi.one_apply",
"set.Ico",
"set.indicator_mul_left",
"set.mem_Ic... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_mul_upcrossings_before_le_integral [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hfN : ∀ ω, a ≤ f N ω) (hfzero : 0 ≤ f 0) (hab : a < b) :
(b - a) * μ[upcrossings_before a b f N] ≤ μ[f N] | calc (b - a) * μ[upcrossings_before a b f N]
≤ μ[∑ k in finset.range N, upcrossing_strat a b f N k * (f (k + 1) - f k)] :
begin
rw ← integral_mul_left,
refine integral_mono_of_nonneg _ ((hf.sum_upcrossing_strat_mul a b N).integrable N) _,
{ exact eventually_of_forall (λ ω, mul_nonneg (sub_nonneg.2 hab.le) (na... | lemma | measure_theory.integral_mul_upcrossings_before_le_integral | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.range",
"nat.cast_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
crossing_pos_eq (hab : a < b) :
upper_crossing_time 0 (b - a) (λ n ω, (f n ω - a)⁺) N n = upper_crossing_time a b f N n ∧
lower_crossing_time 0 (b - a) (λ n ω, (f n ω - a)⁺) N n = lower_crossing_time a b f N n | begin
have hab' : 0 < b - a := sub_pos.2 hab,
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω,
{ intros i ω,
refine ⟨λ h, _, λ h, _⟩,
{ rwa [← sub_le_sub_iff_right a,
← lattice_ordered_comm_group.pos_eq_self_of_pos_pos (lt_of_lt_of_le hab' h)] },
{ rw ← sub_le_sub_iff_right a at h,
rwa... | lemma | measure_theory.crossing_pos_eq | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ih",
"set.mem_Icc",
"set.mem_Ici",
"set.mem_Iic",
"tsub_le_iff_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_before_pos_eq (hab : a < b) :
upcrossings_before 0 (b - a) (λ n ω, (f n ω - a)⁺) N ω = upcrossings_before a b f N ω | by simp_rw [upcrossings_before, (crossing_pos_eq hab).1] | lemma | measure_theory.upcrossings_before_pos_eq | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_upcrossings_before_le_integral_pos_part_aux [is_finite_measure μ]
(hf : submartingale f ℱ μ) (hab : a < b) :
(b - a) * μ[upcrossings_before a b f N] ≤ μ[λ ω, (f N ω - a)⁺] | begin
refine le_trans (le_of_eq _) (integral_mul_upcrossings_before_le_integral
(hf.sub_martingale (martingale_const _ _ _)).pos
(λ ω, lattice_ordered_comm_group.pos_nonneg _)
(λ ω, lattice_ordered_comm_group.pos_nonneg _) (sub_pos.2 hab)),
simp_rw [sub_zero, ← upcrossings_before_pos_eq hab],
refl,
en... | lemma | measure_theory.mul_integral_upcrossings_before_le_integral_pos_part_aux | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.mul_integral_upcrossings_before_le_integral_pos_part [is_finite_measure μ]
(a b : ℝ) (hf : submartingale f ℱ μ) (N : ℕ) :
(b - a) * μ[upcrossings_before a b f N] ≤ μ[λ ω, (f N ω - a)⁺] | begin
by_cases hab : a < b,
{ exact mul_integral_upcrossings_before_le_integral_pos_part_aux hf hab },
{ rw [not_lt, ← sub_nonpos] at hab,
exact le_trans (mul_nonpos_of_nonpos_of_nonneg hab (integral_nonneg (λ ω, nat.cast_nonneg _)))
(integral_nonneg (λ ω, lattice_ordered_comm_group.pos_nonneg _)) }
end | theorem | measure_theory.submartingale.mul_integral_upcrossings_before_le_integral_pos_part | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"mul_nonpos_of_nonpos_of_nonneg",
"nat.cast_nonneg"
] | **Doob's upcrossing estimate**: given a real valued discrete submartingale `f` and real
values `a` and `b`, we have `(b - a) * 𝔼[upcrossings_before a b f N] ≤ 𝔼[(f N - a)⁺]` where
`upcrossings_before a b f N` is the number of times the process `f` crossed from below `a` to above
`b` before the time `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
upcrossings_before_eq_sum (hab : a < b) :
upcrossings_before a b f N ω =
∑ i in finset.Ico 1 (N + 1), {n | upper_crossing_time a b f N n ω < N}.indicator 1 i | begin
by_cases hN : N = 0,
{ simp [hN] },
rw ← finset.sum_Ico_consecutive _ (nat.succ_le_succ zero_le')
(nat.succ_le_succ (upcrossings_before_le f ω hab)),
have h₁ : ∀ k ∈ finset.Ico 1 (upcrossings_before a b f N ω + 1),
{n : ℕ | upper_crossing_time a b f N n ω < N}.indicator 1 k = 1,
{ rintro k hk,
... | lemma | measure_theory.upcrossings_before_eq_sum | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.Ico",
"finset.mem_Ico",
"mul_one",
"mul_zero",
"nat.add_succ_sub_one",
"nat.card_Ico",
"nat.succ_le_iff",
"smul_eq_mul",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.measurable_upcrossings_before (hf : adapted ℱ f) (hab : a < b) :
measurable (upcrossings_before a b f N) | begin
have : upcrossings_before a b f N =
λ ω, ∑ i in finset.Ico 1 (N + 1), {n | upper_crossing_time a b f N n ω < N}.indicator 1 i,
{ ext ω,
exact upcrossings_before_eq_sum hab },
rw this,
exact finset.measurable_sum _ (λ i hi, measurable.indicator measurable_const $
ℱ.le N _ (hf.is_stopping_time_u... | lemma | measure_theory.adapted.measurable_upcrossings_before | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"finset.Ico",
"measurable",
"measurable.indicator",
"measurable_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
adapted.integrable_upcrossings_before [is_finite_measure μ]
(hf : adapted ℱ f) (hab : a < b) :
integrable (λ ω, (upcrossings_before a b f N ω : ℝ)) μ | begin
have : ∀ᵐ ω ∂μ, ‖(upcrossings_before a b f N ω : ℝ)‖ ≤ N,
{ refine eventually_of_forall (λ ω, _),
rw [real.norm_eq_abs, nat.abs_cast, nat.cast_le],
refine upcrossings_before_le _ _ hab },
exact ⟨measurable.ae_strongly_measurable
(measurable_from_top.comp (hf.measurable_upcrossings_before hab)),
... | lemma | measure_theory.adapted.integrable_upcrossings_before | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"nat.abs_cast",
"nat.cast_le",
"real.norm_eq_abs"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings [preorder ι] [order_bot ι] [has_Inf ι]
(a b : ℝ) (f : ι → Ω → ℝ) (ω : Ω) : ℝ≥0∞ | ⨆ N, (upcrossings_before a b f N ω : ℝ≥0∞) | def | measure_theory.upcrossings | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"has_Inf",
"order_bot"
] | The number of upcrossings of a realization of a stochastic process (`upcrossing` takes value
in `ℝ≥0∞` and so is allowed to be `∞`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adapted.measurable_upcrossings (hf : adapted ℱ f) (hab : a < b) :
measurable (upcrossings a b f) | measurable_supr (λ N, measurable_from_top.comp (hf.measurable_upcrossings_before hab)) | lemma | measure_theory.adapted.measurable_upcrossings | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"measurable",
"measurable_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upcrossings_lt_top_iff :
upcrossings a b f ω < ∞ ↔ ∃ k, ∀ N, upcrossings_before a b f N ω ≤ k | begin
have : upcrossings a b f ω < ∞ ↔ ∃ k : ℝ≥0, upcrossings a b f ω ≤ k,
{ split,
{ intro h,
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr,
exact ⟨r, le_rfl⟩ },
{ rintro ⟨k, hk⟩,
exact lt_of_le_of_lt hk ennreal.coe_lt_top } },
simp_rw [this, upcrossings, supr_le_iff],
split; r... | lemma | measure_theory.upcrossings_lt_top_iff | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ennreal.coe_le_coe",
"ennreal.coe_lt_top",
"ennreal.coe_nat",
"exists_nat_ge",
"lift",
"nat.cast_le",
"supr_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part [is_finite_measure μ]
(a b : ℝ) (hf : submartingale f ℱ μ) :
ennreal.of_real (b - a) * ∫⁻ ω, upcrossings a b f ω ∂μ ≤
⨆ N, ∫⁻ ω, ennreal.of_real ((f N ω - a)⁺) ∂μ | begin
by_cases hab : a < b,
{ simp_rw [upcrossings],
have : ∀ N, ∫⁻ ω, ennreal.of_real ((f N ω - a)⁺) ∂μ = ennreal.of_real (∫ ω, (f N ω - a)⁺ ∂μ),
{ intro N,
rw of_real_integral_eq_lintegral_of_real,
{ exact (hf.sub_martingale (martingale_const _ _ _)).pos.integrable _ },
{ exact eventuall... | lemma | measure_theory.submartingale.mul_lintegral_upcrossings_le_lintegral_pos_part | probability.martingale | src/probability/martingale/upcrossing.lean | [
"data.set.intervals.monotone",
"probability.process.hitting_time",
"probability.martingale.basic"
] | [
"ae_measurable",
"ennreal.mul_supr",
"ennreal.of_real",
"ennreal.of_real_le_of_real",
"ennreal.of_real_mul",
"le_supr",
"nat.cast_le",
"nnreal.coe_nat_cast",
"supr_le_iff",
"zero_mul"
] | A variant of Doob's upcrossing estimate obtained by taking the supremum on both sides. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{u} pmf (α : Type u) : Type u | { f : α → ℝ≥0∞ // has_sum f 1 } | def | pmf | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"has_sum"
] | A probability mass function, or discrete probability measures is a function `α → ℝ≥0∞` such
that the values have (infinite) sum `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_like : fun_like (pmf α) α (λ p, ℝ≥0∞) | { coe := λ p a, p.1 a,
coe_injective' := λ p q h, subtype.eq h } | instance | pmf.fun_like | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"fun_like",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {p q : pmf α} (h : ∀ x, p x = q x) : p = q | fun_like.ext p q h | lemma | pmf.ext | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"fun_like.ext",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {p q : pmf α} : p = q ↔ ∀ x, p x = q x | fun_like.ext_iff | lemma | pmf.ext_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"fun_like.ext_iff",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_coe_one (p : pmf α) : has_sum p 1 | p.2 | lemma | pmf.has_sum_coe_one | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"has_sum",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe (p : pmf α) : ∑' a, p a = 1 | p.has_sum_coe_one.tsum_eq | lemma | pmf.tsum_coe | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe_ne_top (p : pmf α) : ∑' a, p a ≠ ∞ | p.tsum_coe.symm ▸ ennreal.one_ne_top | lemma | pmf.tsum_coe_ne_top | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"ennreal.one_ne_top",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_coe_indicator_ne_top (p : pmf α) (s : set α) : ∑' a, s.indicator p a ≠ ∞ | ne_of_lt (lt_of_le_of_lt (tsum_le_tsum (λ a, set.indicator_apply_le (λ _, le_rfl))
ennreal.summable ennreal.summable) (lt_of_le_of_ne le_top p.tsum_coe_ne_top)) | lemma | pmf.tsum_coe_indicator_ne_top | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"ennreal.summable",
"le_rfl",
"le_top",
"pmf",
"tsum_le_tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ne_zero (p : pmf α) : ⇑p ≠ 0 | λ hp, zero_ne_one ((tsum_zero.symm.trans (tsum_congr $
λ x, symm (congr_fun hp x))).trans p.tsum_coe) | lemma | pmf.coe_ne_zero | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf",
"tsum_congr",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support (p : pmf α) : set α | function.support p | def | pmf.support | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"function.support",
"pmf"
] | The support of a `pmf` is the set where it is nonzero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_support_iff (p : pmf α) (a : α) : a ∈ p.support ↔ p a ≠ 0 | iff.rfl | lemma | pmf.mem_support_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_nonempty (p : pmf α) : p.support.nonempty | function.support_nonempty_iff.2 p.coe_ne_zero | lemma | pmf.support_nonempty | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_zero_iff (p : pmf α) (a : α) : p a = 0 ↔ a ∉ p.support | by rw [mem_support_iff, not_not] | lemma | pmf.apply_eq_zero_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"not_not",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_pos_iff (p : pmf α) (a : α) : 0 < p a ↔ a ∈ p.support | pos_iff_ne_zero.trans (p.mem_support_iff a).symm | lemma | pmf.apply_pos_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_one_iff (p : pmf α) (a : α) : p a = 1 ↔ p.support = {a} | begin
refine ⟨λ h, set.subset.antisymm (λ a' ha', by_contra $ λ ha, _) (λ a' ha',
ha'.symm ▸ (p.mem_support_iff a).2 (λ ha, zero_ne_one $ ha.symm.trans h)), λ h, trans
(symm $ tsum_eq_single a (λ a' ha', (p.apply_eq_zero_iff a').2 (h.symm ▸ ha'))) p.tsum_coe⟩,
suffices : 1 < ∑' a, p a,
from ne_of_lt thi... | lemma | pmf.apply_eq_one_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"by_contra",
"ennreal.add_lt_add_of_le_of_lt",
"ennreal.one_ne_top",
"ennreal.summable",
"le_rfl",
"lt_of_le_of_ne'",
"pmf",
"set.subset.antisymm",
"tsum_congr",
"tsum_eq_single",
"tsum_ne_zero_iff",
"zero_le'",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_le_one (p : pmf α) (a : α) : p a ≤ 1 | has_sum_le (by { intro b, split_ifs; simp only [h, zero_le', le_rfl] })
(has_sum_ite_eq a (p a)) (has_sum_coe_one p) | lemma | pmf.coe_le_one | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"has_sum_ite_eq",
"has_sum_le",
"le_rfl",
"pmf",
"zero_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_ne_top (p : pmf α) (a : α) : p a ≠ ∞ | ne_of_lt (lt_of_le_of_lt (p.coe_le_one a) ennreal.one_lt_top) | lemma | pmf.apply_ne_top | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"ennreal.one_lt_top",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_lt_top (p : pmf α) (a : α) : p a < ∞ | lt_of_le_of_ne le_top (p.apply_ne_top a) | lemma | pmf.apply_lt_top | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"le_top",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure (p : pmf α) : outer_measure α | outer_measure.sum (λ (x : α), p x • dirac x) | def | pmf.to_outer_measure | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | Construct an `outer_measure` from a `pmf`, by assigning measure to each set `s : set α` equal
to the sum of `p x` for for each `x ∈ α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_outer_measure_apply : p.to_outer_measure s = ∑' x, s.indicator p x | tsum_congr (λ x, smul_dirac_apply (p x) x s) | lemma | pmf.to_outer_measure_apply | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_caratheodory : p.to_outer_measure.caratheodory = ⊤ | begin
refine (eq_top_iff.2 $ le_trans (le_Inf $ λ x hx, _) (le_sum_caratheodory _)),
exact let ⟨y, hy⟩ := hx in ((le_of_eq (dirac_caratheodory y).symm).trans
(le_smul_caratheodory _ _)).trans (le_of_eq hy),
end | lemma | pmf.to_outer_measure_caratheodory | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"le_Inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_apply_finset (s : finset α) : p.to_outer_measure s = ∑ x in s, p x | begin
refine (to_outer_measure_apply p s).trans ((@tsum_eq_sum _ _ _ _ _ _ s _).trans _),
{ exact λ x hx, set.indicator_of_not_mem hx _ },
{ exact finset.sum_congr rfl (λ x hx, set.indicator_of_mem hx _) }
end | lemma | pmf.to_outer_measure_apply_finset | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"finset",
"tsum_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_apply_singleton (a : α) : p.to_outer_measure {a} = p a | begin
refine (p.to_outer_measure_apply {a}).trans ((tsum_eq_single a $ λ b hb, _).trans _),
{ exact ite_eq_right_iff.2 (λ hb', false.elim $ hb hb') },
{ exact ite_eq_left_iff.2 (λ ha', false.elim $ ha' rfl) }
end | lemma | pmf.to_outer_measure_apply_singleton | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"tsum_eq_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_injective : (to_outer_measure : pmf α → outer_measure α).injective | λ p q h, pmf.ext (λ x, (p.to_outer_measure_apply_singleton x).symm.trans
((congr_fun (congr_arg _ h) _).trans $ q.to_outer_measure_apply_singleton x)) | lemma | pmf.to_outer_measure_injective | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf",
"pmf.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_inj {p q : pmf α} :
p.to_outer_measure = q.to_outer_measure ↔ p = q | to_outer_measure_injective.eq_iff | lemma | pmf.to_outer_measure_inj | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_apply_eq_zero_iff : p.to_outer_measure s = 0 ↔ disjoint p.support s | begin
rw [to_outer_measure_apply, ennreal.tsum_eq_zero],
exact function.funext_iff.symm.trans set.indicator_eq_zero',
end | lemma | pmf.to_outer_measure_apply_eq_zero_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"disjoint",
"ennreal.tsum_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_apply_eq_one_iff : p.to_outer_measure s = 1 ↔ p.support ⊆ s | begin
refine (p.to_outer_measure_apply s).symm ▸ ⟨λ h a hap, _, λ h, _⟩,
{ refine by_contra (λ hs, ne_of_lt _ (h.trans p.tsum_coe.symm)),
have hs' : s.indicator p a = 0 := set.indicator_apply_eq_zero.2 (λ hs', false.elim $ hs hs'),
have hsa : s.indicator p a < p a := hs'.symm ▸ (p.apply_pos_iff a).2 hap,
... | lemma | pmf.to_outer_measure_apply_eq_one_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"by_contra",
"ennreal.tsum_lt_tsum",
"le_rfl",
"set.not_mem_subset",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_apply_inter_support :
p.to_outer_measure (s ∩ p.support) = p.to_outer_measure s | by simp only [to_outer_measure_apply, pmf.support, set.indicator_inter_support] | lemma | pmf.to_outer_measure_apply_inter_support | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf.support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_mono {s t : set α} (h : s ∩ p.support ⊆ t) :
p.to_outer_measure s ≤ p.to_outer_measure t | le_trans (le_of_eq (to_outer_measure_apply_inter_support p s).symm) (p.to_outer_measure.mono h) | lemma | pmf.to_outer_measure_mono | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | Slightly stronger than `outer_measure.mono` having an intersection with `p.support` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_outer_measure_apply_eq_of_inter_support_eq {s t : set α}
(h : s ∩ p.support = t ∩ p.support) : p.to_outer_measure s = p.to_outer_measure t | le_antisymm (p.to_outer_measure_mono (h.symm ▸ (set.inter_subset_left t p.support)))
(p.to_outer_measure_mono (h ▸ (set.inter_subset_left s p.support))) | lemma | pmf.to_outer_measure_apply_eq_of_inter_support_eq | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"set.inter_subset_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_apply_fintype [fintype α] : p.to_outer_measure s = ∑ x, s.indicator p x | (p.to_outer_measure_apply s).trans (tsum_eq_sum (λ x h, absurd (finset.mem_univ x) h)) | lemma | pmf.to_outer_measure_apply_fintype | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"finset.mem_univ",
"fintype",
"tsum_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure [measurable_space α] (p : pmf α) : measure α | p.to_outer_measure.to_measure ((to_outer_measure_caratheodory p).symm ▸ le_top) | def | pmf.to_measure | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"le_top",
"measurable_space",
"pmf"
] | Since every set is Carathéodory-measurable under `pmf.to_outer_measure`,
we can further extend this `outer_measure` to a `measure` on `α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_outer_measure_apply_le_to_measure_apply : p.to_outer_measure s ≤ p.to_measure s | le_to_measure_apply p.to_outer_measure _ s | lemma | pmf.to_outer_measure_apply_le_to_measure_apply | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_eq_to_outer_measure_apply (hs : measurable_set s) :
p.to_measure s = p.to_outer_measure s | to_measure_apply p.to_outer_measure _ hs | lemma | pmf.to_measure_apply_eq_to_outer_measure_apply | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply (hs : measurable_set s) : p.to_measure s = ∑' x, s.indicator p x | (p.to_measure_apply_eq_to_outer_measure_apply s hs).trans (p.to_outer_measure_apply s) | lemma | pmf.to_measure_apply | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_singleton (a : α) (h : measurable_set ({a} : set α)) :
p.to_measure {a} = p a | by simp [to_measure_apply_eq_to_outer_measure_apply _ _ h, to_outer_measure_apply_singleton] | lemma | pmf.to_measure_apply_singleton | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_eq_zero_iff (hs : measurable_set s) :
p.to_measure s = 0 ↔ disjoint p.support s | by rw [to_measure_apply_eq_to_outer_measure_apply p s hs,
to_outer_measure_apply_eq_zero_iff] | lemma | pmf.to_measure_apply_eq_zero_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"disjoint",
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_eq_one_iff (hs : measurable_set s) : p.to_measure s = 1 ↔ p.support ⊆ s | (p.to_measure_apply_eq_to_outer_measure_apply s hs : p.to_measure s = p.to_outer_measure s).symm
▸ (p.to_outer_measure_apply_eq_one_iff s) | lemma | pmf.to_measure_apply_eq_one_iff | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_inter_support (hs : measurable_set s) (hp : measurable_set p.support) :
p.to_measure (s ∩ p.support) = p.to_measure s | by simp [p.to_measure_apply_eq_to_outer_measure_apply s hs,
p.to_measure_apply_eq_to_outer_measure_apply _ (hs.inter hp)] | lemma | pmf.to_measure_apply_inter_support | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_mono {s t : set α} (hs : measurable_set s) (ht : measurable_set t)
(h : s ∩ p.support ⊆ t) : p.to_measure s ≤ p.to_measure t | by simpa only [p.to_measure_apply_eq_to_outer_measure_apply, hs, ht]
using to_outer_measure_mono p h | lemma | pmf.to_measure_mono | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_eq_of_inter_support_eq {s t : set α} (hs : measurable_set s)
(ht : measurable_set t) (h : s ∩ p.support = t ∩ p.support) : p.to_measure s = p.to_measure t | by simpa only [p.to_measure_apply_eq_to_outer_measure_apply, hs, ht]
using to_outer_measure_apply_eq_of_inter_support_eq p h | lemma | pmf.to_measure_apply_eq_of_inter_support_eq | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"measurable_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_injective : (to_measure : pmf α → measure α).injective | λ p q h, pmf.ext (λ x, (p.to_measure_apply_singleton x $ measurable_set_singleton x).symm.trans
((congr_fun (congr_arg _ h) _).trans $ q.to_measure_apply_singleton x $
measurable_set_singleton x)) | lemma | pmf.to_measure_injective | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf",
"pmf.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_inj {p q : pmf α} : p.to_measure = q.to_measure ↔ p = q | to_measure_injective.eq_iff | lemma | pmf.to_measure_inj | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_finset (s : finset α) : p.to_measure s = ∑ x in s, p x | (p.to_measure_apply_eq_to_outer_measure_apply s s.measurable_set).trans
(p.to_outer_measure_apply_finset s) | lemma | pmf.to_measure_apply_finset | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_of_finite (hs : s.finite) : p.to_measure s = ∑' x, s.indicator p x | (p.to_measure_apply_eq_to_outer_measure_apply s hs.measurable_set).trans
(p.to_outer_measure_apply s) | lemma | pmf.to_measure_apply_of_finite | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_apply_fintype [fintype α] : p.to_measure s = ∑ x, s.indicator p x | (p.to_measure_apply_eq_to_outer_measure_apply s s.to_finite.measurable_set).trans
(p.to_outer_measure_apply_fintype s) | lemma | pmf.to_measure_apply_fintype | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pmf [countable α] [measurable_space α] [measurable_singleton_class α]
(μ : measure α) [h : is_probability_measure μ] : pmf α | ⟨λ x, μ ({x} : set α), ennreal.summable.has_sum_iff.2 (trans (symm $
(tsum_indicator_apply_singleton μ set.univ measurable_set.univ).symm.trans
(tsum_congr (λ x, congr_fun (set.indicator_univ _) x))) (h.measure_univ))⟩ | def | measure_theory.measure.to_pmf | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"countable",
"measurable_set.univ",
"measurable_singleton_class",
"measurable_space",
"pmf",
"tsum_congr"
] | Given that `α` is a countable, measurable space with all singleton sets measurable,
we can convert any probability measure into a `pmf`, where the mass of a point
is the measure of the singleton set under the original measure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_pmf_apply (x : α) : μ.to_pmf x = μ {x} | rfl | lemma | measure_theory.measure.to_pmf_apply | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pmf_to_measure : μ.to_pmf.to_measure = μ | measure.ext (λ s hs, by simpa only [μ.to_pmf.to_measure_apply s hs,
← μ.tsum_indicator_apply_singleton s hs]) | lemma | measure_theory.measure.to_pmf_to_measure | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure.is_probability_measure [measurable_space α] (p : pmf α) :
is_probability_measure (p.to_measure) | ⟨by simpa only [measurable_set.univ, to_measure_apply_eq_to_outer_measure_apply,
set.indicator_univ, to_outer_measure_apply, ennreal.coe_eq_one] using tsum_coe p⟩ | instance | pmf.to_measure.is_probability_measure | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"ennreal.coe_eq_one",
"measurable_set.univ",
"measurable_space",
"pmf"
] | The measure associated to a `pmf` by `to_measure` is a probability measure | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_measure_to_pmf : p.to_measure.to_pmf = p | pmf.ext (λ x, by rw [← p.to_measure_apply_singleton x (measurable_set_singleton x),
p.to_measure.to_pmf_apply]) | lemma | pmf.to_measure_to_pmf | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [
"pmf.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_eq_iff_eq_to_pmf (μ : measure α) [is_probability_measure μ] :
p.to_measure = μ ↔ p = μ.to_pmf | by rw [← to_measure_inj, measure.to_pmf_to_measure] | lemma | pmf.to_measure_eq_iff_eq_to_pmf | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pmf_eq_iff_to_measure_eq (μ : measure α) [is_probability_measure μ] :
μ.to_pmf = p ↔ μ = p.to_measure | by rw [← to_measure_inj, measure.to_pmf_to_measure] | lemma | pmf.to_pmf_eq_iff_to_measure_eq | probability.probability_mass_function | src/probability/probability_mass_function/basic.lean | [
"topology.instances.ennreal",
"measure_theory.measure.measure_space"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) (p : pmf α) : pmf β | bind p (pure ∘ f) | def | pmf.map | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"pmf"
] | The functorial action of a function on a `pmf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monad_map_eq_map {α β : Type*} (f : α → β) (p : pmf α) : f <$> p = p.map f | rfl | lemma | pmf.monad_map_eq_map | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_apply : (map f p) b = ∑' a, if b = f a then p a else 0 | by simp [map] | lemma | pmf.map_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_map : (map f p).support = f '' p.support | set.ext (λ b, by simp [map, @eq_comm β b]) | lemma | pmf.support_map | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_map_iff : b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b | by simp | lemma | pmf.mem_support_map_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_pure_comp : bind p (pure ∘ f) = map f p | rfl | lemma | pmf.bind_pure_comp | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map id p = p | bind_pure _ | lemma | pmf.map_id | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (g : β → γ) : (p.map f).map g = p.map (g ∘ f) | by simp [map] | lemma | pmf.map_comp | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pure_map (a : α) : (pure a).map f = pure (f a) | pure_bind _ _ | lemma | pmf.pure_map | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bind (q : α → pmf β) (f : β → γ) :
(p.bind q).map f = p.bind (λ a, (q a).map f) | bind_bind _ _ _ | lemma | pmf.map_bind | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"map_bind",
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bind_map (p : pmf α) (f : α → β) (q : β → pmf γ) :
(p.map f).bind q = p.bind (q ∘ f) | (bind_bind _ _ _).trans (congr_arg _ (funext (λ a, pure_bind _ _))) | lemma | pmf.bind_map | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_const : p.map (function.const α b) = pure b | by simp only [map, bind_const, function.comp_const] | lemma | pmf.map_const | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"function.comp_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_map_apply :
(p.map f).to_outer_measure s = p.to_outer_measure (f ⁻¹' s) | by simp [map, set.indicator, to_outer_measure_apply p (f ⁻¹' s)] | lemma | pmf.to_outer_measure_map_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"set.indicator"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_map_apply [measurable_space α] [measurable_space β] (hf : measurable f)
(hs : measurable_set s) : (p.map f).to_measure s = p.to_measure (f ⁻¹' s) | begin
rw [to_measure_apply_eq_to_outer_measure_apply _ s hs,
to_measure_apply_eq_to_outer_measure_apply _ (f ⁻¹' s) (measurable_set_preimage hf hs)],
exact to_outer_measure_map_apply f p s,
end | lemma | pmf.to_measure_map_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"measurable",
"measurable_set",
"measurable_set_preimage",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq (q : pmf (α → β)) (p : pmf α) : pmf β | q.bind (λ m, p.bind $ λ a, pure (m a)) | def | pmf.seq | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"pmf"
] | The monadic sequencing operation for `pmf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monad_seq_eq_seq {α β : Type*} (q : pmf (α → β)) (p : pmf α) : q <*> p = q.seq p | rfl | lemma | pmf.monad_seq_eq_seq | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"pmf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
seq_apply : (seq q p) b = ∑' (f : α → β) (a : α), if b = f a then q f * p a else 0 | begin
simp only [seq, mul_boole, bind_apply, pure_apply],
refine tsum_congr (λ f, (ennreal.tsum_mul_left).symm.trans (tsum_congr (λ a, _))),
simpa only [mul_zero] using mul_ite (b = f a) (q f) (p a) 0
end | lemma | pmf.seq_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"ennreal.tsum_mul_left",
"mul_boole",
"mul_ite",
"mul_zero",
"tsum_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_seq : (seq q p).support = ⋃ f ∈ q.support, f '' p.support | set.ext (λ b, by simp [-mem_support_iff, seq, @eq_comm β b]) | lemma | pmf.support_seq | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_seq_iff : b ∈ (seq q p).support ↔ ∃ (f ∈ q.support), b ∈ f '' p.support | by simp | lemma | pmf.mem_support_seq_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_finset (f : α → ℝ≥0∞) (s : finset α) (h : ∑ a in s, f a = 1)
(h' : ∀ a ∉ s, f a = 0) : pmf α | ⟨f, h ▸ has_sum_sum_of_ne_finset_zero h'⟩ | def | pmf.of_finset | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"finset",
"has_sum_sum_of_ne_finset_zero",
"pmf"
] | Given a finset `s` and a function `f : α → ℝ≥0∞` with sum `1` on `s`,
such that `f a = 0` for `a ∉ s`, we get a `pmf` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_finset_apply (a : α) : of_finset f s h h' a = f a | rfl | lemma | pmf.of_finset_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_of_finset : (of_finset f s h h').support = s ∩ (function.support f) | set.ext (λ a, by simpa [mem_support_iff] using mt (h' a)) | lemma | pmf.support_of_finset | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"function.support",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_of_finset_iff (a : α) : a ∈ (of_finset f s h h').support ↔ a ∈ s ∧ f a ≠ 0 | by simp | lemma | pmf.mem_support_of_finset_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_finset_apply_of_not_mem {a : α} (ha : a ∉ s) : of_finset f s h h' a = 0 | h' a ha | lemma | pmf.of_finset_apply_of_not_mem | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_of_finset_apply :
(of_finset f s h h').to_outer_measure t = ∑' x, t.indicator f x | to_outer_measure_apply (of_finset f s h h') t | lemma | pmf.to_outer_measure_of_finset_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_measure_of_finset_apply [measurable_space α] (ht : measurable_set t) :
(of_finset f s h h').to_measure t = ∑' x, t.indicator f x | (to_measure_apply_eq_to_outer_measure_apply _ t ht).trans
(to_outer_measure_of_finset_apply h h' t) | lemma | pmf.to_measure_of_finset_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"measurable_set",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_fintype [fintype α] (f : α → ℝ≥0∞) (h : ∑ a, f a = 1) : pmf α | of_finset f finset.univ h (λ a ha, absurd (finset.mem_univ a) ha) | def | pmf.of_fintype | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"finset.mem_univ",
"finset.univ",
"fintype",
"pmf"
] | Given a finite type `α` and a function `f : α → ℝ≥0∞` with sum 1, we get a `pmf`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_fintype_apply (a : α) : of_fintype f h a = f a | rfl | lemma | pmf.of_fintype_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_of_fintype : (of_fintype f h).support = function.support f | rfl | lemma | pmf.support_of_fintype | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [
"function.support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_support_of_fintype_iff (a : α) : a ∈ (of_fintype f h).support ↔ f a ≠ 0 | iff.rfl | lemma | pmf.mem_support_of_fintype_iff | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_outer_measure_of_fintype_apply :
(of_fintype f h).to_outer_measure s = ∑' x, s.indicator f x | to_outer_measure_apply (of_fintype f h) s | lemma | pmf.to_outer_measure_of_fintype_apply | probability.probability_mass_function | src/probability/probability_mass_function/constructions.lean | [
"probability.probability_mass_function.monad"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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